Max-cut parameterized above the edwards-erdős bound

Authors:
Robert Crowston;Mark Jones;Matthias Mnich
Affiliations:
Royal Holloway, University of London, UK;Royal Holloway, University of London, UK;Cluster of Excellence, Saarbrücken, Germany
Venue:
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Year:
2012

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Abstract

We study the boundary of tractability for the Max-Cut problem in graphs. Our main result shows that Max-Cut above the Edwards-Erdős bound is fixed-parameter tractable: we give an algorithm that for any connected graph with n vertices and m edges finds a cut of size $$ \frac{m}{2} + \frac{n-1}{4} + k $$ in time 2O(k)·n4, or decides that no such cut exists. This answers a long-standing open question from parameterized complexity that has been posed a number of times over the past 15 years. Our algorithm is asymptotically optimal, under the Exponential Time Hypothesis, and is strengthened by a polynomial-time computable kernel of polynomial size.